The Spacetime Elastic Model (STEM)

We present the principles of the Spacetime Elastic Model (STEM), a framework in which General Relativity is supplemented by two axioms: the spatial geometry of the Universe is flat on average at all times, and spacetime is a homogeneous medium with a finite, non-zero stiffness characterised by one dimensionless constant alpha. Translating the axioms into four dimensions through deficit angles (holonomies), we show that the ratio of the true to the baryonic deficit angle of a galaxy is the mass-discrepancy factor, and that homogeneity together with the existence of flat rotation curves leads to g = g_N / (1 - exp(-sqrt(g_N/a_0))) This reproduces the known empirical radial acceleration relation (RAR) and the baryonic Tully--Fisher relation; its agreement with the SPARC data is that of the published RAR analyses, since an independent STEM fit has not yet been made. Space around a galaxy is a cone. At low acceleration the temporal and spatial potentials coincide, so lensing and dynamics see the same mass. In divergence form the local law adds no net source: every gravitational basin carries zero net phantom mass, with positive halos balanced by depleted regions between galaxies, so that flatness is preserved whenever backreaction is negligible (e.g.\ at linear order). Homogeneity makes dark energy uniform, and we prove, within the class of classical models without a new dimensionful constant, that a permanent non-zero stiffness in a flat, expanding universe requires Lambda > 0, with a_0 = alpha * c^2 * sqrt(Lambda/3) and alpha ≈ 0.22 In the early Universe the response of the medium is dormant (the primordial ripples are in the strong-field regime), so the CMB sees Einstein's theory alone; toy-level calculations (no full Boltzmann treatment) indicate that a quiet, pressureless component is required. Since the response is zero while the mass is there, the membrane cannot be only a field: it exists, its mass is born with the mass of matter, and a membrane simply sitting at rest is what the CMB calls dark matter; later the same budget is shaped by the stiffness of the membrane. With a conserved dark-matter budget (verified from the CMB to today at the few-per-cent level), uniform dark energy and flatness, the expansion rate becomes the single unknown of one equation, H_0^2 = (8*pi*G/3)*rho_m0 + (a_0/(alpha*c))^2, which links the Hubble tension to a 10\% change in alpha. This is a working note: results are tagged as empirical, derived, hypothetical or open.